Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 41 x^{2} - 117 x^{3} + 169 x^{4}$ |
Frobenius angles: | $\pm0.109149799241$, $\pm0.400911184348$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.122157.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $85$ | $28645$ | $4886905$ | $810796725$ | $137471390800$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $171$ | $2225$ | $28387$ | $370250$ | $4827579$ | $62773205$ | $815828803$ | $10604636345$ | $137858461686$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+x^5+10x^4+4x^3+10x+11$
- $y^2=11x^6+7x^5+12x^4+8x^3+3x^2+6x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The endomorphism algebra of this simple isogeny class is 4.0.122157.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.13.j_bp | $2$ | 2.169.b_adj |